Dynamics of a Birth-Pulse Single-Species Model with Restricted Toxin Input and Pulse Harvesting

We consider a birth-pulses single-species model with restricted toxin input and pulse harvesting in a polluted environment. Pollution accumulates as a slowly decaying stock and is assumed to affect the growth of the renewable resource population. Firstly, by using the discrete dynamical system determined by the stroboscopic map, we obtain an exact 1-period solution of system whose birth function is Ricker function or Beverton-Holt function and obtain the threshold conditions for their stability. Furthermore, we show that the timing of harvesting has a strong impact on the maximum annual sustainable yield. The best timing of harvesting is immediately after the birth pulses. Finally, we investigate the effect of the amount of toxin input on the stable resource population size. We find that when the birth rate is comparatively lower, the population size is decreasing with the increase of toxin input; that when the birth rate is high, the population size may begin to rise and then drop with the increase of toxin input.